Calculus: Analytic Geometry and Calculus, with Vectors

3.4 Continuity^147

The graphs of the signurn and Heaviside functions shown in Figures 3.191
and 3.192 should indicate that these functions are continuous every-
where except at x = 0. One who has seen numerous examples of func-
tions and their graphs should realize that he can enter the construction
business to produce more examples. He can start with a clean coordinate
system and, as in Figures 3.46 and 3.47, mark points ±x1, ±x2, ±x3,

Y

Y=ql(x)

• xi - za

Figure 3.46

on the x axis and then sketch a part of a graph which oscillatesthrough

these points in any way he likes. Provided only that the graph contains

no two different points having the same x coordinate, the graph will be

the graph of a function. In Figure 3.46 the graph is drawn tangent over

and over again to the lines having equations y = 1 and y = -1. In

Figure 3.47 the graph is drawn tangent over and over again to the

Y

Y=q2(x)

Figure 3.47

parabolas having the equations y = x2 and y = -x2. It can be shown

that the graphs of the functions defined by ql(x) = sin(1/x) when x 96 0

and q2(x) = x2 sin(1/x) when x 34 0 and q2(0) = 0 look very much like the

graphs in Figures 3.46 and 3.47, but we need not worry about this matter

now. It should be clear from Figure 3.46 that ql cannot be continuous

at x = 0 because lim ql(x) does not exist. For the function q2 the story-

X 4o

is different. Since

(3.471) -x2 < q2(X) < x2,